Z. Zhang, Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues, Appl. Math. Mech. 35 (2014) 437-452.Zhang Z-Y (2014) Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues. Applied ...
General Solution Of A Matrix with Repeated Eigenvalues: The general solution of the linear homogeneous system X′=AX Which has repeated eigenvalues is Y(t)=c1v1eλ1t+c2(v1teλ1t+v2eλ1t)+c3v3eλ2t To find the values of ...
General Solution Of A Matrix with Repeated Eigenvalues: The general solution of the linear homogeneous system {eq}{X}'=AX{/eq} Which has repeated eigenvalues is {eq}X\left ( t \right )=c_{1}v_{1}e^{\lambda _{1}t}+c_{2}\left (...
Remarkably, if r is a vector of length n, a random sparse matrix with eigenvalues equal to the elements of r is produced. Eigenvalues are discussed in Section 2.15. A positive definite matrix has all its eigenvalues positive and consequently such a matrix can be generated by choosing each of...
A matrix can have repeated eigenvalues and be diagonalizable, as diagonal matrices with repeated diagonal entries show. What is needed for diagonalizability is that every -times repeated eigenvalue has linearly independent eigenvectors associated with it. Equivalently, the algebraic and geometric multiplic...
However, a similarity transformation may be used to diagonalize A if it does not have any repeated eigenvalues [13]. If we diagonalize the state transition matrix A, then the update of the states will require 2×L multiplications and L additions. This can result in substantial computational ...
However, for matrices the nonuniqueness is complicated by the presence of repeated eigenvalues. For example, the matrix is an upper triangular logarithm of the identity matrix for any , whereas the obvious logarithms are the diagonal matrices , for integers , , and . Notice that the repeated ...
eigenvectors and eigenvalues. Methods forcalculating them based on dense, in-place factorizations are intractably expensive forlarge matrices [59, 60]. Iterative methods involving repeated matrix–vector multipli-cations [34, 17, 54] of f er reduced computational and memory costs, particularly for...
NumPy的主要对象是同种元素的多维数组。这是一个所有的元素都是一种类型、通过一个正整数元组索引的元素表格(通常是元素是数字)。 在NumPy中维度(dimensions)叫做轴(axes),轴的个数叫做秩(rank,但是和线性代数中的秩不是一样的,在用python求线代中的秩中,我们用numpy包中的linalg.matrix_rank方法计算矩阵的秩,...
If the system does not have repeated eigenvalues, the mode shape matrix is a full rank matrix. The rank of a matrix product does not exceed the rank of any individual matrix. 2.1.5 Similarity of matrices Two real matrices [A] and [B] are similar if there exists a non-singular matrix ...